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A235488 Squarefree numbers which yield zero when their prime factors are xored together.

%I A235488 #20 Apr 01 2017 10:15:19
%S A235488 70,646,1798,2145,3526,5865,6006,9177,11305,13110,16422,20553,20806,
%T A235488 21489,23529,28905,28985,30305,31465,37961,38086,38454,42441,44022,
%U A235488 44998,45353,45942,46345,53985,54230,55913,60630,60697,61705,62049,64790,78406,80934,81158
%N A235488 Squarefree numbers which yield zero when their prime factors are xored together.
%C A235488 All n for which A008683(n) <> 0 and A072594(n) = 0.
%C A235488 It seems that an analogous case as A072595 for GF(2)[X]-polynomials is just the squares of GF(2)[X]-polynomials (A000695), thus in that ring, the sequence analogous to this one would be empty.
%C A235488 This sequence happens also to encode in the prime factorization of n a certain subset of the Nim game positions that are second-player win.
%H A235488 Antti Karttunen and Charles R Greathouse IV, <a href="/A235488/b235488.txt">Table of n, a(n) for n = 1..10000</a> (first 75 terms from Karttunen)
%H A235488 Wikipedia, <a href="http://en.wikipedia.org/wiki/Nim">Nim</a>
%e A235488 70 is included, as 70 = 2*5*7, whose binary representations are '10', '101' and '111', which when all are xored (cf. A003987) together, cancel all 1-bits, thus yielding zero.
%e A235488 212585 is included, as 212585 = 5*17*41*61, and when we xor their base-2 representations together:
%e A235488      101
%e A235488    10001
%e A235488   101001
%e A235488   111101
%e A235488 --------
%e A235488   000000
%e A235488 we get only zeros, because in each column (bit-position), there is an even number of 1-bits.
%t A235488 Select[Range[82000],SquareFreeQ[#]&&BitXor@@FactorInteger[#][[All,1]]==0&] (* _Harvey P. Dale_, Apr 01 2017 *)
%o A235488 (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o A235488 (define A235488 (MATCHING-POS 1 1 (lambda (n) (and (not (zero? (A008683 n))) (zero? (A072594 n))))))
%o A235488 (PARI) is(n)=if(n<9, return(0)); my(f=factor(n)); vecmax(f[,2])==1 && fold(bitxor, f[,1])==0 \\ _Charles R Greathouse IV_, Aug 06 2016
%Y A235488 Intersection of A005117 and A072595 (equally: of A005117 and A072596).
%Y A235488 Cf. A003987, A235490, A079599, A048833, A050314.
%K A235488 nonn,base
%O A235488 1,1
%A A235488 _Antti Karttunen_, Jan 22 2014